Merge branch 'master' of https://github.com/AndreasFMueller/SeminarMatrizen

84 files changed, 2093 insertions, 0 deletions

diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
new file mode 100644
index 0000000..7afeea1
--- /dev/null
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -0,0 +1,22 @@
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 =								\
+	../slides/7/symmetrien.tex					\
+	../slides/7/algebraisch.tex					\
+	../slides/7/parameter.tex					\
+	../slides/7/mannigfaltigkeit.tex				\
+	../slides/7/sl2.tex						\
+	../slides/7/drehung.tex						\
+	../slides/7/drehanim.tex					\
+	../slides/7/semi.tex						\
+	../slides/7/kurven.tex						\
+	../slides/7/einparameter.tex					\
+	../slides/7/ableitung.tex					\
+	../slides/7/liealgebra.tex					\
+	../slides/7/kommutator.tex					\
+	../slides/7/dg.tex						\
+	../slides/7/chapter.tex
+
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
new file mode 100644
index 0000000..12f9084
--- /dev/null
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -0,0 +1,68 @@
+%
+% ableitung.tex -- Ableitung in der Lie-Gruppe
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ableitung in der Matrix-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{O}(n)$}
+\uncover<2->{%
+$s \mapsto A(s)\in\operatorname{O}(n)$
+}
+\begin{align*}
+\uncover<3->{I
+&=
+A(s)^tA(s)}
+\\
+\uncover<4->{0
+=
+\frac{d}{ds} I
+&=
+\frac{d}{ds} (A(s)^t A(s))}
+\\
+&\uncover<5->{=
+\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)}
+\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}}
+\uncover<7->{0
+&=
+\dot{A}(0)^t
++
+\dot{A}(0)}
+\\
+\uncover<8->{\Leftrightarrow
+\qquad
+\dot{A}(0)^t &= -\dot{A}(0)}
+\end{align*}
+\uncover<9->{%
+``Tangentialvektoren'' sind antisymmetrische Matrizen
+}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
+\uncover<2->{%
+$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
+}
+\begin{align*}
+\uncover<3->{1 &= \det A(t)}
+\\
+\uncover<10->{0
+=
+\frac{d}{dt}1
+&=
+\frac{d}{dt} \det A(t)}
+\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}}
+\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)}
+\end{align*}
+\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex
new file mode 100644
index 0000000..31d209a
--- /dev/null
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -0,0 +1,115 @@
+%
+% algebraisch.tex -- algebraische Definition der Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erhaltungsgrössen und Algebra}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Längen und Winkel}
+Längenmessung mit Skalarprodukt
+\begin{align*}
+\|\vec{v}\|^2
+&=
+\langle \vec{v},\vec{v}\rangle
+=
+\vec{v}\cdot \vec{v}
+\uncover<2->{=
+\vec{v}^t\vec{v}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Flächeninhalt/Volumen}
+$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
+\\
+Volumen des Parallelepipeds: $\det V$
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-7pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Längenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\vec{x}^t\vec{y}
+&=
+(A\vec{x})
+\cdot
+(A\vec{y})
+\uncover<5->{=
+(A\vec{x})^t
+(A\vec{y})}
+\\
+\uncover<6->{
+\vec{x}^tI\vec{y}
+&=
+\vec{x}^tA^tA\vec{y}}
+\uncover<7->{
+\Rightarrow I=A^tA}
+\end{align*}
+\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<9->{%
+\begin{block}{Volumenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\det(V)
+&=
+\det(AV)
+\uncover<10->{=
+\det(A)\det(V)}
+\\
+\uncover<11->{
+1&=\det(A)}
+\end{align*}
+\uncover<10->{
+(Produktsatz für Determinante)
+}
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-3pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Orthogonale Matrizen}
+Längentreue Abbildungen = orthogonale Matrizen:
+\[
+O(n)
+=
+\{
+A \in \operatorname{GL}_n(\mathbb{R})
+\;|\;
+A^tA=I
+\}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<13->{%
+\begin{block}{``Spezielle'' Matrizen}
+Volumen-/Orientierungserhaltende Transformationen:
+\[
+\operatorname{SL}_n(\mathbb R) 
+=
+\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
new file mode 100644
index 0000000..079cf16
--- /dev/null
+++ b/vorlesungen/slides/7/chapter.tex
@@ -0,0 +1,19 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{7/symmetrien.tex}
+\folie{7/algebraisch.tex}
+\folie{7/parameter.tex}
+\folie{7/mannigfaltigkeit.tex}
+\folie{7/sl2.tex}
+\folie{7/drehung.tex}
+\folie{7/drehanim.tex}
+\folie{7/semi.tex}
+\folie{7/kurven.tex}
+\folie{7/einparameter.tex}
+\folie{7/ableitung.tex}
+\folie{7/liealgebra.tex}
+\folie{7/kommutator.tex}
+\folie{7/dg.tex}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex
new file mode 100644
index 0000000..4447bac
--- /dev/null
+++ b/vorlesungen/slides/7/dg.tex
@@ -0,0 +1,92 @@
+%
+% dg.tex -- Differentialgleichung für die Exponentialabbildung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zurück zur Lie-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
+Ableitung von $\gamma(t)$ an der Stelle $t$:
+\begin{align*}
+\dot{\gamma}(t)
+&\uncover<2->{=
+\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
+}
+\\
+&\uncover<3->{=
+\frac{d}{ds}
+\gamma(t+s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<4->{=
+\frac{d}{ds}
+\gamma(t)\gamma(s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<5->{=
+\gamma(t)
+\frac{d}{ds}
+\gamma(s)
+\bigg|_{s=0}
+}
+\uncover<6->{=
+\gamma(t) \dot{\gamma}(0)
+}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Differentialgleichung}
+\vspace{-10pt}
+\[
+\dot{\gamma}(t) = \gamma(t) A
+\quad
+\text{mit}
+\quad
+A=\dot{\gamma}(0)\in LG
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<8->{%
+\begin{block}{Lösung}
+Exponentialfunktion
+\[
+\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<9->{%
+\begin{block}{Kontrolle: Tangentialvektor berechnen}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}e^{At}
+&\uncover<10->{=
+\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!}
+}
+\\
+&\uncover<11->{=
+\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
+}
+\\
+&\uncover<12->{=
+\sum_{k=0} A^k\frac{t^k}{k!}
+A
+}
+\uncover<13->{=
+e^{At} A
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex
new file mode 100644
index 0000000..ac136f1
--- /dev/null
+++ b/vorlesungen/slides/7/drehanim.tex
@@ -0,0 +1,155 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
+
+\makeatletter
+\hoffset=-2cm
+\advance\textwidth2cm
+\hsize\textwidth
+\columnwidth\textwidth
+\makeatother
+
+\begin{frame}[t,plain]
+\vspace{-5pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=white] (-4,-4) rectangle (9,4.5);
+
+\def\a{60}
+
+\pgfmathparse{tan(\a)}
+\xdef\T{\pgfmathresult}
+
+\pgfmathparse{-sin(\a)*cos(\a)}
+\xdef\S{\pgfmathresult}
+
+\pgfmathparse{1/cos(\a)}
+\xdef\E{\pgfmathresult}
+
+\def\N{20}
+\pgfmathparse{2*\N}
+\xdef\Nzwei{\pgfmathresult}
+\pgfmathparse{3*\N}
+\xdef\Ndrei{\pgfmathresult}
+
+\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
+\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
+\begin{itemize}
+\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
+as shear mappings and axis rescalings
+\[
+A=
+\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
+\begin{pmatrix}1&s\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\t&1\end{pmatrix}
+\]
+\item Most rotations can be decomposed into a product of
+shear mappings and axis rescalings
+\end{itemize}
+\end{block}
+\end{minipage}};
+
+\foreach \d in {1,2,...,\Ndrei}{
+	% Scherung in Y-Richtung
+	\ifnum \d>\N
+		\pgfmathparse{\T}
+	\else
+		\pgfmathparse{\T*(\d-1)/(\N-1)}
+	\fi
+	\xdef\t{\pgfmathresult}
+
+	% Scherung in X-Richtung
+	\ifnum \d>\Nzwei
+		\xdef\s{\S}
+	\else
+		\ifnum \d<\N
+			\xdef\s{0}
+		\else
+			\ifnum \d=\N
+				\xdef\s{0}
+			\else
+				\pgfmathparse{\S*(\d-\N-1)/(\N-1)}
+				\xdef\s{\pgfmathresult}
+			\fi
+		\fi
+	\fi
+
+	% Reskalierung der Achsen
+	\ifnum \d>\Nzwei
+		\pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
+	\else
+		\pgfmathparse{1}
+	\fi
+	\xdef\e{\pgfmathresult}
+
+	% Matrixelemente 
+	\pgfmathparse{(\e)*((\s)*(\t)+1)}
+	\xdef\A{\pgfmathresult}
+
+	\pgfmathparse{(\e)*(\s)}
+	\xdef\B{\pgfmathresult}
+
+	\pgfmathparse{(\t)/(\e)}
+	\xdef\C{\pgfmathresult}
+
+	\pgfmathparse{1/(\e)}
+	\xdef\D{\pgfmathresult}
+
+	\only<\d>{
+		\node at (5.0,-0.9) [below right] {$
+			\begin{aligned}
+				t &= \t                 \\
+				s &= \s                 \\
+				d &= \e                 \\
+				D &= \begin{pmatrix}
+					\A&\B\\
+					\C&\D
+				     \end{pmatrix}
+					\qquad
+					\only<60>{\checkmark}
+			\end{aligned}
+		$};
+	}
+
+	\begin{scope}
+		\clip (-4.05,-4.05) rectangle (4.05,4.05);
+		\only<\d>{
+			\foreach \x in {-6,...,6}{
+				\draw[color=blue,line width=0.5pt]
+					\punkt{\x}{-12} -- \punkt{\x}{12};
+			}
+			\foreach \y in {-12,...,12}{
+				\draw[color=darkgreen,line width=0.5pt]
+					\punkt{-6}{\y} -- \punkt{6}{\y};
+			}
+
+			\foreach \r in {1,2,3,4}{
+				\draw[color=red] plot[domain=0:359,samples=360]
+					({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
+					--
+					cycle;
+			}
+		}
+	\end{scope}
+
+%	\uncover<\d>{
+%		\node at (5,4) {\d};
+%	}
+}
+
+\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
+\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
new file mode 100644
index 0000000..2d7b317
--- /dev/null
+++ b/vorlesungen/slides/7/drehung.tex
@@ -0,0 +1,132 @@
+%
+% drehung.tex -- Drehung aus streckungen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung aus Streckungen und Scherungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Drehung}
+{\color{blue}Längen}, {\color<2->{blue}Winkel},
+{\color<2->{darkgreen}Orientierung}
+erhalten
+\uncover<2->{
+\[
+\operatorname{SO}(2)
+=
+{\color{blue}\operatorname{O}(2)}
+\cap
+{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
+\]}
+\vspace{-20pt}
+\end{block}
+\uncover<3->{%
+\begin{block}{Zusammensetzung}
+Eine Drehung muss als Zusammensetzung geschrieben werden können:
+\[
+D_{\alpha} 
+=
+\begin{pmatrix}
+\cos\alpha & -\sin\alpha\\
+\sin\alpha &\phantom{-}\cos\alpha
+\end{pmatrix}
+=
+DST
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<12->{%
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
+}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\begin{block}{Ansatz}
+\vspace{-12pt}
+\begin{align*}
+DST
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+1&-s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&0\\
+t&1
+\end{pmatrix}
+\\
+&\uncover<5->{=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+1-st&-s\\
+   t& 1
+\end{pmatrix}
+}
+\\
+&\uncover<6->{=
+\begin{pmatrix}
+{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
+{\color<9->{blue}ct}&{\color<8->{red}c}
+\end{pmatrix}}
+\uncover<7->{=
+\begin{pmatrix}
+{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
+{\color<9->{blue}\sin\alpha} & \phantom{-}  {\color<8->{red}\cos\alpha}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Koeffizientenvergleich}
+\vspace{-15pt}
+\begin{align*}
+\uncover<8->{
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }}
+&&
+&
+\uncover<9->{
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
+\uncover<10->{
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+}
+\\
+\uncover<11->{
+{\color{orange} (1-st)c^{-t}}
+&=
+\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
new file mode 100644
index 0000000..5171085
--- /dev/null
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -0,0 +1,93 @@
+%
+% einparameter.tex -- Einparameter Untergruppen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Einparameter-Untergruppen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
+die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit}
+\[
+\uncover<3->{
+\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
+}
+\]
+\end{block}
+\uncover<4->{%
+\begin{block}{Drehungen}
+Drehmatrizen bilden Einparameter- Untergruppen
+\begin{align*}
+t \mapsto D_{x,t}
+&=
+\begin{pmatrix}
+1&0&0\\
+0&\cos t&-\sin t\\
+0&\sin t& \cos t
+\end{pmatrix}
+\\
+D_{x,t}D_{x,s}
+&=
+D_{x,t+s}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+1&s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&t\\
+0&1
+\end{pmatrix}
+=
+\begin{pmatrix}
+1&s+t\\
+0&1
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<6->{%
+\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+e^s&0\\0&e^{-s}
+\end{pmatrix}
+\begin{pmatrix}
+e^t&0\\0&e^{-t}
+\end{pmatrix}
+=
+\begin{pmatrix}
+e^{t+s}&0\\0&e^{-(t+s)}
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<7->{%
+\begin{block}{Gemischt}
+\vspace{-12pt}
+\begin{gather*}
+A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
+\\
+\text{dank}\quad
+\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
+=I
+\end{gather*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
new file mode 100644
index 0000000..cc67c8a
--- /dev/null
+++ b/vorlesungen/slides/7/images/Makefile
@@ -0,0 +1,19 @@
+#
+# Makefile -- Illustrationen zu 
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+# 
+all:	rodriguez.jpg 
+
+rodriguez.png:	rodriguez.pov
+	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
+
+rodriguez.jpg:	rodriguez.png 
+	convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
+
+commutator:	commutator.ini commutator.pov common.inc
+	povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
+jpg:
+	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
+
+
diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg
new file mode 100644
index 0000000..b2dbdb2
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c01.jpg
diff --git a/vorlesungen/slides/7/images/c/c02.jpg b/vorlesungen/slides/7/images/c/c02.jpg
new file mode 100644
index 0000000..9b45ba3
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c02.jpg
diff --git a/vorlesungen/slides/7/images/c/c03.jpg b/vorlesungen/slides/7/images/c/c03.jpg
new file mode 100644
index 0000000..882be40
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c03.jpg
diff --git a/vorlesungen/slides/7/images/c/c04.jpg b/vorlesungen/slides/7/images/c/c04.jpg
new file mode 100644
index 0000000..5d26572
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c04.jpg
diff --git a/vorlesungen/slides/7/images/c/c05.jpg b/vorlesungen/slides/7/images/c/c05.jpg
new file mode 100644
index 0000000..458f565
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c05.jpg
diff --git a/vorlesungen/slides/7/images/c/c06.jpg b/vorlesungen/slides/7/images/c/c06.jpg
new file mode 100644
index 0000000..cd40cda
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c06.jpg
diff --git a/vorlesungen/slides/7/images/c/c07.jpg b/vorlesungen/slides/7/images/c/c07.jpg
new file mode 100644
index 0000000..3349795
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c07.jpg
diff --git a/vorlesungen/slides/7/images/c/c08.jpg b/vorlesungen/slides/7/images/c/c08.jpg
new file mode 100644
index 0000000..65048cf
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c08.jpg
diff --git a/vorlesungen/slides/7/images/c/c09.jpg b/vorlesungen/slides/7/images/c/c09.jpg
new file mode 100644
index 0000000..000d502
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c09.jpg
diff --git a/vorlesungen/slides/7/images/c/c10.jpg b/vorlesungen/slides/7/images/c/c10.jpg
new file mode 100644
index 0000000..064c96b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c10.jpg
diff --git a/vorlesungen/slides/7/images/c/c11.jpg b/vorlesungen/slides/7/images/c/c11.jpg
new file mode 100644
index 0000000..c67bc5d
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c11.jpg
diff --git a/vorlesungen/slides/7/images/c/c12.jpg b/vorlesungen/slides/7/images/c/c12.jpg
new file mode 100644
index 0000000..f2174de
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c12.jpg
diff --git a/vorlesungen/slides/7/images/c/c13.jpg b/vorlesungen/slides/7/images/c/c13.jpg
new file mode 100644
index 0000000..3cb5ae6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c13.jpg
diff --git a/vorlesungen/slides/7/images/c/c14.jpg b/vorlesungen/slides/7/images/c/c14.jpg
new file mode 100644
index 0000000..6985e4f
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c14.jpg
diff --git a/vorlesungen/slides/7/images/c/c15.jpg b/vorlesungen/slides/7/images/c/c15.jpg
new file mode 100644
index 0000000..5bb5aed
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c15.jpg
diff --git a/vorlesungen/slides/7/images/c/c16.jpg b/vorlesungen/slides/7/images/c/c16.jpg
new file mode 100644
index 0000000..cc77fa1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c16.jpg
diff --git a/vorlesungen/slides/7/images/c/c17.jpg b/vorlesungen/slides/7/images/c/c17.jpg
new file mode 100644
index 0000000..a60e2e9
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c17.jpg
diff --git a/vorlesungen/slides/7/images/c/c18.jpg b/vorlesungen/slides/7/images/c/c18.jpg
new file mode 100644
index 0000000..1d08c29
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c18.jpg
diff --git a/vorlesungen/slides/7/images/c/c19.jpg b/vorlesungen/slides/7/images/c/c19.jpg
new file mode 100644
index 0000000..9210c60
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c19.jpg
diff --git a/vorlesungen/slides/7/images/c/c20.jpg b/vorlesungen/slides/7/images/c/c20.jpg
new file mode 100644
index 0000000..8951883
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c20.jpg
diff --git a/vorlesungen/slides/7/images/c/c21.jpg b/vorlesungen/slides/7/images/c/c21.jpg
new file mode 100644
index 0000000..5169c7e
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c21.jpg
diff --git a/vorlesungen/slides/7/images/c/c22.jpg b/vorlesungen/slides/7/images/c/c22.jpg
new file mode 100644
index 0000000..bdeb90b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c22.jpg
diff --git a/vorlesungen/slides/7/images/c/c23.jpg b/vorlesungen/slides/7/images/c/c23.jpg
new file mode 100644
index 0000000..fa3eac7
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c23.jpg
diff --git a/vorlesungen/slides/7/images/c/c24.jpg b/vorlesungen/slides/7/images/c/c24.jpg
new file mode 100644
index 0000000..52adc13
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c24.jpg
diff --git a/vorlesungen/slides/7/images/c/c25.jpg b/vorlesungen/slides/7/images/c/c25.jpg
new file mode 100644
index 0000000..d557497
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c25.jpg
diff --git a/vorlesungen/slides/7/images/c/c26.jpg b/vorlesungen/slides/7/images/c/c26.jpg
new file mode 100644
index 0000000..f825f49
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c26.jpg
diff --git a/vorlesungen/slides/7/images/c/c27.jpg b/vorlesungen/slides/7/images/c/c27.jpg
new file mode 100644
index 0000000..0de6c3e
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c27.jpg
diff --git a/vorlesungen/slides/7/images/c/c28.jpg b/vorlesungen/slides/7/images/c/c28.jpg
new file mode 100644
index 0000000..d9f89a4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c28.jpg
diff --git a/vorlesungen/slides/7/images/c/c29.jpg b/vorlesungen/slides/7/images/c/c29.jpg
new file mode 100644
index 0000000..937f692
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c29.jpg
diff --git a/vorlesungen/slides/7/images/c/c30.jpg b/vorlesungen/slides/7/images/c/c30.jpg
new file mode 100644
index 0000000..45901cc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c30.jpg
diff --git a/vorlesungen/slides/7/images/c/c31.jpg b/vorlesungen/slides/7/images/c/c31.jpg
new file mode 100644
index 0000000..eb52bec
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c31.jpg
diff --git a/vorlesungen/slides/7/images/c/c32.jpg b/vorlesungen/slides/7/images/c/c32.jpg
new file mode 100644
index 0000000..a011640
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c32.jpg
diff --git a/vorlesungen/slides/7/images/c/c33.jpg b/vorlesungen/slides/7/images/c/c33.jpg
new file mode 100644
index 0000000..ecbb8bd
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c33.jpg
diff --git a/vorlesungen/slides/7/images/c/c34.jpg b/vorlesungen/slides/7/images/c/c34.jpg
new file mode 100644
index 0000000..8a624d1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c34.jpg
diff --git a/vorlesungen/slides/7/images/c/c35.jpg b/vorlesungen/slides/7/images/c/c35.jpg
new file mode 100644
index 0000000..33765a1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c35.jpg
diff --git a/vorlesungen/slides/7/images/c/c36.jpg b/vorlesungen/slides/7/images/c/c36.jpg
new file mode 100644
index 0000000..e38a448
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c36.jpg
diff --git a/vorlesungen/slides/7/images/c/c37.jpg b/vorlesungen/slides/7/images/c/c37.jpg
new file mode 100644
index 0000000..9f823f4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c37.jpg
diff --git a/vorlesungen/slides/7/images/c/c38.jpg b/vorlesungen/slides/7/images/c/c38.jpg
new file mode 100644
index 0000000..19c96fc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c38.jpg
diff --git a/vorlesungen/slides/7/images/c/c39.jpg b/vorlesungen/slides/7/images/c/c39.jpg
new file mode 100644
index 0000000..c00482b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c39.jpg
diff --git a/vorlesungen/slides/7/images/c/c40.jpg b/vorlesungen/slides/7/images/c/c40.jpg
new file mode 100644
index 0000000..de9fca4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c40.jpg
diff --git a/vorlesungen/slides/7/images/c/c41.jpg b/vorlesungen/slides/7/images/c/c41.jpg
new file mode 100644
index 0000000..1b8a3d1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c41.jpg
diff --git a/vorlesungen/slides/7/images/c/c42.jpg b/vorlesungen/slides/7/images/c/c42.jpg
new file mode 100644
index 0000000..6d4e5bc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c42.jpg
diff --git a/vorlesungen/slides/7/images/c/c43.jpg b/vorlesungen/slides/7/images/c/c43.jpg
new file mode 100644
index 0000000..b39551f
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c43.jpg
diff --git a/vorlesungen/slides/7/images/c/c44.jpg b/vorlesungen/slides/7/images/c/c44.jpg
new file mode 100644
index 0000000..0fe56f6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c44.jpg
diff --git a/vorlesungen/slides/7/images/c/c45.jpg b/vorlesungen/slides/7/images/c/c45.jpg
new file mode 100644
index 0000000..9196ad3
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c45.jpg
diff --git a/vorlesungen/slides/7/images/c/c46.jpg b/vorlesungen/slides/7/images/c/c46.jpg
new file mode 100644
index 0000000..a4f5823
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c46.jpg
diff --git a/vorlesungen/slides/7/images/c/c47.jpg b/vorlesungen/slides/7/images/c/c47.jpg
new file mode 100644
index 0000000..18474dd
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c47.jpg
diff --git a/vorlesungen/slides/7/images/c/c48.jpg b/vorlesungen/slides/7/images/c/c48.jpg
new file mode 100644
index 0000000..d839014
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c48.jpg
diff --git a/vorlesungen/slides/7/images/c/c49.jpg b/vorlesungen/slides/7/images/c/c49.jpg
new file mode 100644
index 0000000..3cb5cb0
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c49.jpg
diff --git a/vorlesungen/slides/7/images/c/c50.jpg b/vorlesungen/slides/7/images/c/c50.jpg
new file mode 100644
index 0000000..de32f8b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c50.jpg
diff --git a/vorlesungen/slides/7/images/c/c51.jpg b/vorlesungen/slides/7/images/c/c51.jpg
new file mode 100644
index 0000000..4bbc224
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c51.jpg
diff --git a/vorlesungen/slides/7/images/c/c52.jpg b/vorlesungen/slides/7/images/c/c52.jpg
new file mode 100644
index 0000000..ebf1db6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c52.jpg
diff --git a/vorlesungen/slides/7/images/c/c53.jpg b/vorlesungen/slides/7/images/c/c53.jpg
new file mode 100644
index 0000000..2ece537
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c53.jpg
diff --git a/vorlesungen/slides/7/images/c/c54.jpg b/vorlesungen/slides/7/images/c/c54.jpg
new file mode 100644
index 0000000..0ffbac2
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c54.jpg
diff --git a/vorlesungen/slides/7/images/c/c55.jpg b/vorlesungen/slides/7/images/c/c55.jpg
new file mode 100644
index 0000000..5f75419
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c55.jpg
diff --git a/vorlesungen/slides/7/images/c/c56.jpg b/vorlesungen/slides/7/images/c/c56.jpg
new file mode 100644
index 0000000..5c0f9ae
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c56.jpg
diff --git a/vorlesungen/slides/7/images/c/c57.jpg b/vorlesungen/slides/7/images/c/c57.jpg
new file mode 100644
index 0000000..9b61179
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c57.jpg
diff --git a/vorlesungen/slides/7/images/c/c58.jpg b/vorlesungen/slides/7/images/c/c58.jpg
new file mode 100644
index 0000000..81d2173
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c58.jpg
diff --git a/vorlesungen/slides/7/images/c/c59.jpg b/vorlesungen/slides/7/images/c/c59.jpg
new file mode 100644
index 0000000..0ae9bac
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c59.jpg
diff --git a/vorlesungen/slides/7/images/c/c60.jpg b/vorlesungen/slides/7/images/c/c60.jpg
new file mode 100644
index 0000000..2bc77e7
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c60.jpg
diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc
new file mode 100644
index 0000000..0e27c9a
--- /dev/null
+++ b/vorlesungen/slides/7/images/common.inc
@@ -0,0 +1,70 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.025;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+	location <18, 15, -50>
+	look_at <0.0, 0.5, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-40, 30, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare l = 1.2;
+
+arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)
+arrow(<  0,  0, -l >, < 0, 0, l >, at, White)
+arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)
+
diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini
new file mode 100644
index 0000000..8c2211e
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -0,0 +1,8 @@
+Input_File_Name=commutator.pov
+Initial_Frame=1
+Final_Frame=60
+Initial_Clock=1
+Final_Clock=60
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m
new file mode 100644
index 0000000..5a448db
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -0,0 +1,111 @@
+#
+# commutator.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+X = [
+	0, 0,  0;
+	0, 0, -1;
+	0, 1,  0
+];
+
+Y = [
+	 0, 0, 1;
+	 0, 0, 0;
+	-1, 0, 0
+];
+
+Z = [
+	0, -1, 0;
+	1,  0, 0;
+	0,  0, 0
+];
+
+function retval = Dx(alpha)
+	retval = [
+		1, 0,           0         ;
+		0, cos(alpha), -sin(alpha);
+		0, sin(alpha),  cos(alpha)
+	];
+end
+
+function retval = Dy(beta)
+	retval = [
+		 cos(beta), 0, sin(beta);
+		 0,         1, 0        ;
+		-sin(beta), 0, cos(beta)
+	];
+end
+
+t = 0.9;
+P = Dx(t) * Dy(t)
+Q = Dy(t) * Dx(t)
+P - Q
+(P - Q) * [0;0;1]
+
+function	retval = kurven(filename, t)
+	retval = -1;
+	N = 20;
+	fn = fopen(filename, "w");
+	fprintf(fn, "//\n");
+	fprintf(fn, "// %s\n", filename);
+	fprintf(fn, "//\n");
+	fprintf(fn, "#macro XYkurve()\n");
+	for i = (0:N-1)
+		v1 = Dx(t * i / N) * [0;0;1];
+		v2 = Dx(t * (i+1) / N) * [0;0;1];
+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1));
+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+	end
+	for i = (0:N-1)
+		v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
+		v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1));
+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+	end
+	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+		v2(1,1), v2(3,1), v2(2,1));
+	fprintf(fn, "#end\n");
+	fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
+		v2(1,1), v2(3,1), v2(2,1));
+	fprintf(fn, "#macro YXkurve()\n");
+	for i = (0:N-1)
+		v1 = Dy(t * i / N) * [0;0;1];
+		v2 = Dy(t * (i+1) / N) * [0;0;1];
+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1));
+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+	end
+	for i = (0:N-1)
+		v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
+		v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1));
+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+	end
+	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+		v2(1,1), v2(3,1), v2(2,1));
+	fprintf(fn, "#end\n");
+	fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
+		v2(1,1), v2(3,1), v2(2,1));
+
+	fclose(fn);
+	retval = 0;
+end
+
+function	retval = kurve(i)
+	n = pi / 180;
+	filename = sprintf("f/%04d.inc", i);
+	kurven(filename, n * i);
+end
+
+for i = (1:60)
+	kurve(i);
+end
diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov
new file mode 100644
index 0000000..9ae11b9
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -0,0 +1,59 @@
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+sphere { O, 0.99
+	pigment {
+		color rgbt<1,1,1,0.5>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#declare filename = concat("f/", str(clock, -4, 0), ".inc");
+
+#include filename
+
+#declare n1 = vcross(<0,1,0>, finalXY);
+#declare n2 = vcross(<0,1,0>, finalYX);
+
+intersection {
+	sphere { O, 1 }
+	plane { -n1, 0 }
+	plane { n2, 0 }
+	pigment {
+		color rgb<0,0.4,0.1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+        XYkurve()
+        pigment {
+                color Red
+        }
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+        YXkurve()
+        pigment {
+                color Blue
+        }
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg
new file mode 100644
index 0000000..5c49700
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.jpg
diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png
new file mode 100644
index 0000000..6d9e9e4
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.png
diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
new file mode 100644
index 0000000..07aec19
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -0,0 +1,118 @@
+//
+// rodriguez.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.020;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+	location <8, 15, -50>
+	look_at <0.1, 0.475, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-4, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#declare K = vnormalize(<0.2,1,0.1>);
+#declare X = vnormalize(<1.1,1,-1.2>);
+#declare O = <0,0,0>;
+
+#declare r = vlength(vcross(K, X)) / vlength(K);
+
+#declare l = 1.0;
+
+arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)
+arrow(<  0,  0, -l >, < 0, 0, l >, at, White)
+arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)
+
+arrow(O, X, at, Red)
+arrow(O, K, at, Blue)
+
+#macro punkt(H,phi)
+	((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
+#end
+
+cone { vdot(K, X) * K, r, O, 0
+	pigment {
+		color rgbt<0.6,0.6,0.6,0.5>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+
+union {
+	#declare phistep = pi / 100;
+	#declare phi = 0;
+	#while (phi < 2 * pi - phistep/2) 
+		sphere { punkt(K, phi), at/2 }
+		cylinder {
+			punkt(X, phi),
+			punkt(X, phi + phistep),
+			at/2
+		}
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color Orange
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
+#declare Darkgreen = rgb<0,0.5,0>;
+arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
new file mode 100644
index 0000000..84bf034
--- /dev/null
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -0,0 +1,166 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kommutator in $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\t{14.0cm}
+\ifthenelse{\boolean{presentation}}{
+\only<1>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};}
+\only<2>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};}
+\only<3>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};}
+\only<4>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};}
+\only<5>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};}
+\only<6>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};}
+\only<7>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};}
+\only<8>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};}
+\only<9>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};}
+\only<10>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};}
+\only<11>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};}
+\only<12>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};}
+\only<13>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};}
+\only<14>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};}
+\only<15>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};}
+\only<16>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};}
+\only<17>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};}
+\only<18>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};}
+\only<19>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};}
+\only<20>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};}
+\only<21>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};}
+\only<22>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};}
+\only<23>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};}
+\only<24>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};}
+\only<25>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};}
+\only<26>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};}
+\only<27>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};}
+\only<28>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};}
+\only<29>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};}
+\only<30>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};}
+\only<31>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};}
+\only<32>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};}
+\only<33>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};}
+\only<34>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};}
+\only<35>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};}
+\only<36>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};}
+\only<37>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};}
+\only<38>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};}
+\only<39>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c39.jpg}};}
+\only<40>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};}
+\only<41>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c41.jpg}};}
+\only<42>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};}
+\only<43>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};}
+\only<44>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};}
+\only<45>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};}
+\only<46>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};}
+\only<47>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c47.jpg}};}
+\only<48>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};}
+\only<49>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};}
+\only<50>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c50.jpg}};}
+\only<51>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};}
+\only<52>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};}
+\only<53>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};}
+\only<54>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};}
+\only<55>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};}
+\only<56>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};}
+\only<57>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};}
+\only<58>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};}
+\only<59>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}
+}{}
+\only<60>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};}
+\coordinate (A) at (-0.3,3);
+\coordinate (B) at (-1.1,2);
+\coordinate (C) at (-2.1,-1.2);
+\draw[->,color=red,line width=1.4pt]
+	(A)
+	to[out=-143,in=60]
+	(B)
+	to[out=-120,in=80]
+	(C);
+%\fill[color=red] (B) circle[radius=0.08];
+\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$};
+\coordinate (D) at (0.3,3.2);
+\coordinate (E) at (1.8,2.8);
+\coordinate (F) at (5.2,-0.3);
+\draw[->,color=blue,line width=1.4pt]
+	(D)
+	to[out=-10,in=157]
+	(E)
+	to[out=-23,in=120]
+	(F);
+\fill[color=blue] (E) circle[radius=0.08];
+\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$};
+\draw[->,color=darkgreen,line width=1.4pt]
+	(0.7,-3.1) to[out=1,in=-160] (3.9,-2.6);
+\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$};
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex
new file mode 100644
index 0000000..e0690eb
--- /dev/null
+++ b/vorlesungen/slides/7/kurven.tex
@@ -0,0 +1,104 @@
+%
+% kurven.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kurven und Tangenten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Kurven}
+Kurve in $\mathbb{R}^n$:
+\vspace{-12pt}
+\[
+\gamma
+\colon
+I=[a,b] \to \mathbb{R}^n
+:
+t\mapsto \gamma(t)
+\uncover<2->{
+=
+\begin{pmatrix}
+x_1(t)\\
+x_2(t)\\
+\vdots\\
+x_n(t)
+\end{pmatrix}
+}
+\]
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (1,0.5);
+\coordinate (B) at (4,0.5);
+\coordinate (C) at (2,2.2);
+\coordinate (D) at (5,2);
+\coordinate (E) at ($(C)+(80:2)$);
+
+\draw[color=red,line width=1.4pt]
+	(A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
+\fill[color=red] (C) circle[radius=0.06];
+\node[color=red] at (C) [left] {$\gamma(t)$};
+
+\uncover<4->{
+	\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
+	\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
+}
+
+\uncover<2->{
+	\draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
+	\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Tangenten}
+Ableitung
+\[
+\frac{d}{dt}\gamma(t)
+=
+\dot{\gamma}(t)
+=
+\begin{pmatrix}
+\dot{x}_1(t)\\
+\dot{x}_2(t)\\
+\vdots\\
+\dot{x}_n(t)
+\end{pmatrix}
+\]
+\uncover<5->{%
+Lineare Approximation:
+\[
+\gamma(t+h)
+=
+\gamma(t)
++
+\dot{\gamma}(t) \cdot h
++
+o(h)
+\]}%
+\vspace{-10pt}
+\begin{itemize}
+\item<6->
+Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
+in $\mathbb{R}^n$ liegen
+\item<7->
+Gilt auch für
+\[
+\operatorname{GL}_n(\mathbb{R})
+\uncover<8->{\subset M_n(\mathbb{R})}
+\uncover<9->{ = \mathbb{R}^{n\times n}}
+\]
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
new file mode 100644
index 0000000..574467b
--- /dev/null
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -0,0 +1,85 @@
+%
+% liealgebra.tex -- Lie-Algebra
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra}
+\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
+\begin{block}{Vektorraum}
+Tangentialvektoren im Punkt $I$:
+\begin{center}
+\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
+\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
+\hline
+\uncover<2->{
+\operatorname{GL}_n(\mathbb{R})
+& beliebige Matrizen
+& M_n(\mathbb{R})
+}
+\\
+\uncover<3->{
+\operatorname{O(n)}
+& antisymmetrische Matrizen
+& \operatorname{o}(n)
+}
+\\
+\uncover<4->{
+\operatorname{SL}_n(\mathbb{R})
+& spurlose Matrizen
+& \operatorname{sl}_2(\mathbb{R})
+}
+\\
+\uncover<5->{
+\operatorname{U(n)}
+& antihermitesche Matrizen
+& \operatorname{u}(n)
+}
+\\
+\uncover<6->{
+\operatorname{SU(n)}
+& spurlose, antihermitesche Matrizen
+& \operatorname{su}(n)
+}
+\end{tabular}
+\end{center}
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\uncover<7->{%
+\begin{block}{Lie-Klammer}
+Kommutator: $[A,B] = AB-BA$
+\end{block}}
+\uncover<8->{%
+\begin{block}{Nachprüfen}
+$[A,B]\in LG$
+für $A,B\in LG$
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<9->{%
+\begin{block}{Algebraische Eigenschaften}
+\begin{itemize}
+\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
+\item<11-> Jacobi-Identität
+\[
+[A,[B,C]]+
+[B,[C,A]]+
+[C,[A,B]]
+= 0
+\]
+\end{itemize}
+\vspace{-13pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}
+Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex
new file mode 100644
index 0000000..077dc9d
--- /dev/null
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -0,0 +1,46 @@
+%
+% mannigfaltigkeit.tex -- Mannigfaltigkeit
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Mannigfaltigkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
+\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
+\[
+\varphi_\beta\circ\varphi_\alpha^{-1}
+\colon
+\varphi_\alpha(U_\alpha\cap U_\beta)
+\to
+\varphi_\beta(U_\alpha\cap U_\beta)
+\]
+\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
+\end{itemize}
+\end{block}
+\vspace{-7pt}
+\uncover<5->{%
+\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
+Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
+\end{block}}
+\vspace{-3pt}
+\uncover<6->{%
+\begin{block}{Lie-Gruppe}
+Gruppe und Mannigfaltigkeit
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex
new file mode 100644
index 0000000..52c8e4a
--- /dev/null
+++ b/vorlesungen/slides/7/parameter.tex
@@ -0,0 +1,107 @@
+%
+% parameter.tex -- Parametrisierung der Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkyellow}{rgb}{1,0.8,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehungen Parametrisieren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.4\textwidth}
+\begin{block}{Drehung um Achsen}
+\vspace{-12pt}
+\begin{align*}
+\uncover<2->{
+D_{x,\alpha}
+&=
+\begin{pmatrix}
+1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
+\end{pmatrix}
+}
+\\
+\uncover<3->{
+D_{y,\beta}
+&=
+\begin{pmatrix}
+\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta
+\end{pmatrix}
+}
+\\
+\uncover<4->{
+D_{z,\gamma}
+&=
+\begin{pmatrix}
+\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
+\end{pmatrix}
+}
+\intertext{\uncover<5->{beliebige Drehung:}}
+\uncover<5->{
+D
+&=
+D_{x,\alpha}
+D_{y,\beta}
+D_{z,\gamma}
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<6->{%
+\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional}
+\uncover<7->{%
+$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
+}
+\\
+\uncover<8->{%
+$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
+}
+\[
+\uncover<9->{
+{\color{red}\vec{x}}
+\mapsto
+}
+\uncover<10->{
+({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
+\cos\omega
++
+}
+\uncover<11->{
+({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
++
+}
+\uncover<9->{
+{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) 
+}
+\]
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+	\node at (0,0)
+		{\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}};
+	\node[color=red] at (1.6,-0.9) {$\vec{x}$};
+	\node[color=blue] at (0.5,2) {$\vec{k}$};
+}
+\uncover<11->{
+	\node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$};
+}
+\uncover<10->{
+	\node[color=yellow] at (2.2,-0.2)
+		{$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$};
+}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\uncover<5->{%
+{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
+dreidimensionale Gruppe}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex
new file mode 100644
index 0000000..66b8d27
--- /dev/null
+++ b/vorlesungen/slides/7/semi.tex
@@ -0,0 +1,117 @@
+%
+% semi.tex -- Beispiele: semidirekte Produkte
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung/Skalierung und Verschiebung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalierung und Verschiebung}
+Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
+\\
+Wirkung auf $\mathbb{R}$:
+\[
+x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Drehung und Verschiebung}
+Gruppe
+$G=
+\{ (\alpha,\vec{t})
+\;|\;
+\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
+\}$
+Wirkung auf $\mathbb{R}^2$:
+\[
+\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(e^{s_1},t_1)(e^{s_2},t_2)x
+&\uncover<4->{=
+(e^{s_1},t_1)(e^{s_2}x+t_2)}
+\\
+&\uncover<5->{=
+e^{s_1+s_2}x + e^{s_1}t_2+t_1}
+\\
+\uncover<6->{
+(e^{s_1},t_1)(e^{s_2},t_2)
+&=
+(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+\vec{x}
+&\uncover<8->{=
+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
+\\
+&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
+\\
+\uncover<10->{
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+&=
+(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(e^s,t) =
+\begin{pmatrix}
+e^s&t\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}x\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(\alpha,\vec{t}) =
+\begin{pmatrix}
+D_{\alpha}&\vec{t}\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}\vec{x}\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
new file mode 100644
index 0000000..a65b4f6
--- /dev/null
+++ b/vorlesungen/slides/7/sl2.tex
@@ -0,0 +1,242 @@
+%
+% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t,fragile]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Determinante}
+\[
+A=\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\;\Rightarrow\;
+\det A = ad-bc
+\]
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Dimension}
+\[
+4\; \text{Variablen}
+-
+1\; \text{Bedingung}
+=
+3\; \text{Dimensionen}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{columns}[t,onlytextwidth]
+\def\s{0.94}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\fill[color=blue!20] (-1,-1) rectangle (1,1);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\ifthenelse{\boolean{presentation}}{
+	\foreach \d in {4,...,10}{
+		\only<\d>{
+			\pgfmathparse{1+(\d-4)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				({-\t},{-1/\t}) rectangle (\t,{1/\t});
+			\foreach \x in {-2,...,2}{
+				\draw[color=red,line width=0.3pt]
+					({\x*\t},-3) -- ({\x*\t},3);
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+				(-3,{\y/\t}) -- (3,{\y/\t});
+			}
+		}
+	}
+	}{}
+	\uncover<11->{
+		\xdef\t{1.6}
+		\fill[color=red!40,opacity=0.5]
+			({-\t},{-1/\t}) rectangle (\t,{1/\t});
+		\foreach \x in {-2,...,2}{
+			\draw[color=red,line width=0.3pt]
+				({\x*\t},-3) -- ({\x*\t},3);
+		}
+		\foreach \y in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+			(-3,{\y/\t}) -- (3,{\y/\t});
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<3->{%
+	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	D
+	=
+	\begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\ifthenelse{\boolean{presentation}}{
+	\foreach \d in {11,...,17}{
+		\only<\d>{
+			\pgfmathparse{(\d-11)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				({-1+\t*(-1)},{-1})
+				--
+				({1+\t*(-1)},{-1})
+				--
+				({1+\t},{1})
+				--
+				({-1+\t},{1})
+				-- cycle;
+			\foreach \x in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+			}
+		}
+	}
+	}{}
+	\uncover<18->{
+		\xdef\t{0.6}
+		\fill[color=red!40,opacity=0.5]
+			({-1+\t*(-1)},{-1})
+			--
+			({1+\t*(-1)},{-1})
+			--
+			({1+\t},{1})
+			--
+			({-1+\t},{1})
+			-- cycle;
+		\foreach \x in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+		}
+		\foreach \y in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<11->{
+	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	S
+	=
+	\begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\ifthenelse{\boolean{presentation}}{
+	\foreach \d in {18,...,24}{
+		\only<\d>{
+			\pgfmathparse{(\d-18)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				(-1,{\t*(-1)-1})
+				--
+				(1,{\t*1-1})
+				--
+				(1,{\t*1+1})
+				--
+				(-1,{\t*(-1)+1})
+				-- cycle;
+			\foreach \x in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+			}
+		}
+	}
+	}{}
+	\uncover<25->{
+		\xdef\t{0.6}
+		\fill[color=red!40,opacity=0.5]
+			(-1,{\t*(-1)-1})
+			--
+			(1,{\t*1-1})
+			--
+			(1,{\t*1+1})
+			--
+			(-1,{\t*(-1)+1})
+			-- cycle;
+		\foreach \x in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+		}
+		\foreach \y in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<18->{%
+\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	T
+	=
+	\begin{pmatrix} 1&0\\t&1\end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
new file mode 100644
index 0000000..35d62d8
--- /dev/null
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -0,0 +1,145 @@
+%
+% symmetrien.tex -- Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Symmetrien}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diskrete Symmetrien}
+\begin{itemize}
+\item<2->
+Ebenen-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+{\tiny
+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\uncover<4->{\!,\;
+\vec{x}
+\mapsto
+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
+}
+\]
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{10}
+\def\b{50}
+\def\r{2}
+\coordinate (O) at (0,0);
+\coordinate (A) at (\b:\r);
+\coordinate (B) at ({180+2*\a-\b}:\r);
+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
+\coordinate (N) at (\a:2);
+\coordinate (D) at (\a:{\r*cos(\b-\a)});
+\uncover<3->{
+\clip (-2.5,-0.45) rectangle (2.5,1.95);
+
+	\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
+		-- cycle;
+	\draw[->,color=darkgreen] (O) -- (N);
+	\node[color=darkgreen] at (N) [above] {$\vec{n}$};
+
+
+	\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
+		-- cycle;
+	\fill[color=red] (O) circle[radius=0.06];
+	\draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
+	\fill[color=blue] (C) circle[radius=0.06];
+	\draw[color=blue,line width=0.1pt] (A) -- (D);
+	\node[color=darkgreen] at (D) [below,rotate=\a]
+		{$(\vec{n}\cdot\vec{x})\vec{n}$};
+	\draw[color=blue,line width=0.5pt] (A)--(B);
+
+	\node[color=blue] at (A) [above right] {$\vec{x}$};
+	\node[color=blue] at (B) [above left] {$\vec{x}'$};
+
+	\node[color=red] at (O) [below left] {$O$};
+
+	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
+	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-5pt}
+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
+\item<5->
+Punkt-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+-
+{\tiny
+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Kontinuierliche Symmetrien}
+\begin{itemize}
+\item<7-> Translation:
+\(
+\vec{x} \mapsto \vec{x} + \vec{t}
+\)
+\item<8-> Drehung:
+\vspace{-3pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{25}
+\def\r{1.3}
+\coordinate (O) at (0,0);
+\begin{scope}
+\clip (-1.1,-0.1) rectangle (2.3,2.3);
+\draw[color=red] (O) circle[radius=2];
+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
+\node at ({0.5*\a}:1) {$\alpha$};
+\node at ({90+0.5*\a}:1) {$\alpha$};
+\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
+\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
+\end{scope}
+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
+\end{tikzpicture}
+\end{center}
+\[
+\uncover<9->{%
+\begin{pmatrix}x\\y\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
+\end{pmatrix}
+\begin{pmatrix}x\\y\end{pmatrix}
+}
+\]
+\end{itemize}
+\end{block}}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Definition}
+Längen/Winkel bleiben erhalten
+\\
+\uncover<11->{%
+$\Rightarrow$ $\exists$ Erhaltungsgrösse}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup